LETTER
doi:10.1038/nature11672
The root of branching river networks
J. Taylor Perron1, Paul W. Richardson1, Ken L. Ferrier1 & Mathieu Lapôtre2
Branching river networks are one of the most widespread and
recognizable features of Earth’s landscapes and have also been
discovered elsewhere in the Solar System1,2. But the mechanisms
that create these patterns and control their spatial scales are poorly
understood. Theories based on probability3–5 or optimality3,6–8
have proven useful9, but do not explain how river networks develop
over time through erosion and sediment transport. Here we show
that branching at the uppermost reaches of river networks is rooted
in two coupled instabilities: first, valleys widen at the expense of
their smaller neighbours, and second, side slopes of the widening
valleys become susceptible to channel incision. Each instability
occurs at a critical ratio of the characteristic timescales for soil
transport and channel incision. Measurements from two field sites
demonstrate that our theory correctly predicts the size of the smallest valleys with tributaries. We also show that the dominant control on the scale of landscape dissection in these sites is the strength
of channel incision, which correlates with aridity and rock weakness, rather than the strength of soil transport. These results imply
that the fine-scale structure of branching river networks is an organized signature of erosional mechanics, not a consequence of random topology.
Observers have long recognized that branching river networks, in
which tributaries merge in the downstream direction, are a fundamental outcome of erosion by flowing water. Early field observations led to conceptual models in which hierarchical drainage
networks result from progressive integration of initially separate drainages through divide migration and stream capture10,11. More than a
century of study of erosional mechanics has culminated in numerical
models that produce landscapes dissected by branching river networks12–14 and lend some support to the early conceptual models,
but a physically based theory that explains the form of tributary
networks has proven elusive15.
One of the main reasons for this shortcoming is that there is no clear
consensus about what such a theory should be able to predict. Early
efforts to identify unique geometric characteristics of natural river
networks11,16,17 failed when the proposed scaling laws were subsequently shown to apply to all hierarchical networks, riverine or otherwise3,4,18. This fuelled a broader class of studies based on principles
other than erosional mechanics, including probabilistic models of
topologically random networks3–5 and optimality arguments based
on energy dissipation or entropy3,6–8. Such studies have provided new
insights into the structure of natural river networks9, but cannot
directly relate river network form to the erosional mechanisms that
shape the topography: we know what the skeleton of a landscape looks
like, but not how it grows.
Thus, despite progress in characterizing the geometry of drainage
networks and modelling landscape evolution, it remains unclear how
the form of drainage networks records the dominant factors that shape
landscapes, such as bedrock properties, tectonic deformation, climate
and life. Here we use a simplified model to show that branching tributary networks form through two coupled instabilities, and propose that
the scale of the smallest valleys with tributaries is a signature of this
process. We then present field measurements from two sites with
drainage networks that differ considerably in scale, and show that both
are consistent with our theory. This comparison reveals how the spatial
structure of river networks records fundamental geological and environmental factors such as rock type and rainfall.
To identify characteristic scales in tributary networks, we mapped
the valley networks in two landscapes with similar erosion processes
but different scales of fluvial dissection (Fig. 1). The Allegheny Plateau
a
Allegheny Plateau
1 km
n = 6,721
103
104
n = 1,376
105
106
b
Gabilan Mesa
First
Second
n = 27,383
order
n = 5,275
Figure 1 | Properties of the study
sites. a, The Allegheny Plateau,
Pennsylvania; b, Gabilan Mesa,
California. Left, area distributions of
first- and second-order drainage
basins, with inset sketches showing
examples of first- and second-order
basins. Coloured lines are histograms
with counts weighted by drainage
basin area to compensate for the
greater abundance of smaller basins.
Insets at centre, shaded relief maps of
small portions of the study sites
generated from laser altimetry, with
vegetation filtered out. Right,
photographs showing views from the
tops of ridgelines, with first-order
valleys in the near distance. See
Methods for data sources.
1 km
103
104
105
106
Drainage basin area (m2)
1
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2École et Observatoire des Sciences de la Terre, Université de
Strasbourg, F-67084 Strasbourg, France.
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LETTER RESEARCH
a
0.5
A
L
Stable
0.4
b
0.3
A/L2
in southwest Pennsylvania and northern West Virginia consists of
horizontally bedded Permian sandstone with minor lenses of limestone, siltstone, shale and coal19. Land cover is dominantly pasture
and deciduous forest, and mean annual precipitation is 105 cm.
Gabilan Mesa is an elevated area on the eastern margin of California’s Salinas Valley where rivers have incised into nearly horizontally bedded Pliocene and Pleistocene conglomerate, sandstone and
siltstone20. The Mesa is an oak savanna that receives 32 cm of rain
per year. Both landscapes are shaped dominantly by soil creep due
to bioturbation, particularly animal burrowing, and by river channel
incision that, in the small drainage basins studied here, is episodic21,
occurring at estimated intervals of years to many decades.
We mapped the valley network over a large area of each landscape
(Methods). We then applied the Horton-Strahler stream ordering
scheme11,16 to identify first-order drainage basins (those with no tributaries) and second-order drainage basins (those with at least one firstorder tributary) (Fig. 1). The upstream areas drained by basins of a
given order are log-normally distributed around a well-defined modal
value (Fig. 1). Moreover, we find that the modal drainage areas of firstand second-order basins in the Allegheny Plateau and Gabilan Mesa
differ considerably. The smallest basins with tributaries are typically
four times larger in the Allegheny Plateau, despite the similar appearance of the landscape (Fig. 1). We seek an explanation for this fundamental scale difference.
Valleys bounded by smooth ridgelines emerge in soil-mantled
landscapes from a competition between river channel incision, which
amplifies topographic perturbations, and soil creep, which damps
perturbations. The extent of valley incision can also be limited by a
threshold for runoff production or surface erosion22, but we make the
simplification that soil creep is the dominant effect, an assumption
supported by previous analyses21,23. The smoothing effect of soil
creep can be characterized with a diffusion time, tdiff 5 L2/D, where
L is a horizontal length scale and D is the soil transport coefficient
(in m2 yr21). Channel incision can be characterized by an advection
time that describes the rate at which changes in elevation propagate
through the drainage network, tadv 5 L122m/K, where m is a constant
and K is a channel incision rate coefficient (in m122m yr21). The ratio
tdiff/tadv, which describes the strength of channel incision relative
to soil creep at a chosen scale L, is analogous to a Péclet number,
Pe 5 KL112m/D (refs 21, 23).
We suggest that branching valley networks will develop at a finer
scale in landscapes where channel incision is stronger relative to soil
creep. To test this idea, we used a landscape evolution model
(Methods) to simulate the development of a ridgeline bounded by
two incising channels for many different values of Pe. Each simulation
formed drainage basins extending from the bounding channels up
towards the drainage divide (Supplementary Fig. 5). The length scale
L was chosen to be the half-width of the ridgeline, which roughly
equals the length of the largest basins (Fig. 2a, inset). (Throughout this
study, L is taken to be the length of a drainage basin, or the length of a
slope on which a drainage basin may develop.) From the final, equilibrium topography in each experiment, we measured the drainage
areas of the basins. A plot of normalized drainage area versus Pe for
all the simulations (Fig. 2a) reveals a transition at 250 , Pe , 300. For
Pe , 250, drainage basins have a uniform size. For Pe . 300, the distribution of basin size is bimodal, with each basin either extending all
the way from a lowering boundary to the central drainage divide, or
extending only part way to the divide and occupying a small space
between larger basins. Nearly all of the larger basins for Pe . 300
developed tributaries (Supplementary Fig. 5).
We interpret this transition in landscape form in terms of a stability
diagram (Fig. 2a). At Pe , 250, an array of parallel, uniformly sized
basins is a stable configuration: additional numerical experiments
confirmed that if any one basin in such a configuration is perturbed
by increasing its drainage area, the topographic divides separating the
enlarged basin from its neighbours will migrate back towards the
c
Unstable
Stable
L
0.2
Probability density
0.1 10
100
1,000
Stable
100
150
200
250
300
350
400
450
Pe
b
c
Time
Figure 2 | Branching instability in valley networks. a, Stability diagram for
first- and second-order drainage basins as a function of Péclet number, Pe.
Insets illustrate the definitions of ridgeline half-width, L, and basin area, A.
Background colour (see key) indicates probability density distribution of
drainage basins generated in numerical experiments, which partly reflects
variable initial conditions (Methods). Paths with blue points and arrows trace
the evolution of the numerical experiments in b and c, which illustrate the
response of an array of parallel valleys to a perturbation in which one valley
(marked with an arrow) is enlarged slightly. Colours in b and c denote drainage
area, with blues corresponding to ridgelines and reds to valleys.
centre of the enlarged basin, such that it shrinks back to the size of
its neighbours (Fig. 2b). At Pe . 300, an array of parallel, uniformly
sized basins is unstable: enlarging the drainage area of any one basin
causes the topographic divides to continue to migrate away from the
centre of the enlarged basin, such that it widens at the expense of its
neighbours (Fig. 2c). This instability propagates through the landscape
until all drainage basins have either grown or shrunk to reach one of
the two stable sizes. Supplementary Fig. 3 shows several drainage
basins in Gabilan Mesa that may have experienced this instability.
This transformation of the drainage network is comparable to an effect
observed in laboratory analogue experiments24, and may explain the
uniform aspect ratios of basins that drain to linear boundaries25.
The qualitative explanation for the critical value of Pe concerns the
feedbacks that operate when a basin is enlarged. The increase in drainage area increases channel discharge, and therefore channel incision
rate. Faster incision further deepens the basin, driving the surrounding
ridgelines towards neighbouring basins and enlarging the drainage
area—a positive feedback. Competing against this is a negative feedback in which deeper valleys with a sharper ‘‘V’’ shape are filled in
faster by diffusive soil creep. For Pe . ,300, the positive feedback is
stronger.
This instability explains the enlargement of some of the basins, but it
does not explain why the enlarged basins develop tributaries on their
side slopes (Fig. 2c). To determine why the tributaries grow, we performed a second experiment in which we subjected an inclined, planar
surface with a prescribed Péclet number (intended to mimic the side
slopes of a drainage basin) to a series of small-amplitude perturbations
with a range of wavelengths, and measured the growth or decay rate of
the perturbations (Fig. 3). The length scale L used to calculate Pe was
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RESEARCH LETTER
networks in both the Allegheny Plateau and Gabilan Mesa are consistent with our proposed mechanism for tributary network development, despite their difference in spatial scale.
This result implies that the ratio of coefficients describing the longterm rates of channel incision and soil creep, K/D, can be estimated
from a drainage basin’s stream order and size. Because m is typically
,0.5 (ref. 28, Supplementary Table 2), Pe < KL2/D as a rule of thumb.
Thus, K/D < Pen/L2, where Pen is the modal Péclet number for a basin
of order n (from Fig. 4, for example, Pe2 < 600). With this approach, it
may be possible to use remote imagery to help calibrate long-term
erosion laws, a result with promise for both terrestrial and planetary
landscapes. The specific expressions presented here only apply to
low-order, steady-state drainage basins shaped by regolith creep and
ephemeral channel incision. Nonetheless, the principle that stream
order and basin size provide a proxy for long-term process rates
may also apply to higher-order drainage basins and landscapes shaped
by different hillslope and channel erosion mechanisms. We show in
the Supplementary Information that the branching mechanism documented in Figs 2 and 3 occurs even with different erosion and transport laws, provided that channel incision depends on drainage area
and regolith transport depends on slope.
To understand how the geological, climatic and biological characteristics of a landscape control the scale of the drainage network, we
performed an additional calculation. Gabilan Mesa is made of weaker
rocks than the Allegheny Plateau, receives less rainfall in a more seasonal distribution, and has vegetation dominated by grasses rather
than forest; it also has larger K/D (Supplementary Table 2) and therefore a finer-scale drainage network (Fig. 1). We determined K and D
independently to discover whether this larger ratio reflects stronger
channel incision, less mobile soil, or both. Combining long-term erosion rates inferred from cosmogenic 10Be in river sediment (Methods,
Supplementary Table 1) with surveys of hillslope topography, we calculated D for the Allegheny Plateau and Gabilan Mesa, and divided D
by D/K to obtain K (Methods, Supplementary Table 2). This calculation clearly shows that the main difference between the sites is the
strength of channel incision: whereas D differs by only 25% (and
is actually larger at Gabilan Mesa, the opposite of what would be
expected if D were responsible for the scale difference), the channel
incision rate factor for a reference drainage area, KAm
ref , is roughly
seven times larger at Gabilan Mesa (Supplementary Table 2). The
magnitudes and uniformity of the soil diffusivities measured here
are consistent with measurements from other landscapes in Mediterranean to humid climates29, and probably reflect similar soil mechanical properties and bioturbation intensities. There are two likely and
chosen to be the horizontal length of this sloping surface, which is the
length of the incipient valleys. For Pe , ,60, perturbations of all
wavelengths decay, and no tributary valleys form. At Pe < 60, wavelengths of roughly one-third of the slope length become unstable, and
grow into incipient tributary valleys. The range of unstable wavelengths widens as Pe increases above this critical value. A similar
wavelength selection has been inferred in analytical studies of incipient
channelization under sheet flow26,27.
To relate this instability to tributary valleys like those in Fig. 2c, we
calculate Péclet numbers for the side slopes of drainage basins with and
without tributaries in the numerical model solutions, using the horizontal lengths of the basin side slopes as L (Fig. 3 inset). This comparison confirms that the difference between valleys with and without
tributaries is whether Pe for their side slopes is greater than or less
than the critical value of ,60. For example, the side slopes of the firstorder basins in Fig. 2b have Pe 5 12, whereas those of the second-order
basins in Fig. 2c have Pe 5 75. Thus, the basin-widening instability
shown in Fig. 2 is accompanied by the formation of tributary valleys
because the lengthening side slopes exceed the critical Pe for growth of
incipient valleys.
Together, these two instabilities provide an explanation for the
characteristic branching pattern of fine-scale tributary networks. In
addition, our theory makes a testable prediction: second-order drainage basins should have Péclet numbers that exceed the critical value of
,300 (Fig. 2a), regardless of the absolute spatial scale of the landscape.
To test this prediction, we calculated Pe for drainage basins in the
Allegheny Plateau and Gabilan Mesa. This requires measurements
of K/D, m and drainage basin length, L, each of which can be estimated
from topographic data. K/D and m have been estimated21 for representative sites in each landscape from an independent analysis of the
topography (Supplementary Table 2). Combining these values with
our measurements of L (Methods), we compiled frequency distributions of Pe (Fig. 4). Unlike the drainage area distributions in Fig. 1,
which differ between sites by a factor of four, the Pe distributions are
quite similar. The critical range of 250 , Pe , 300 (Fig. 2) falls
between the modal values for first- and second-order basins in both
landscapes, consistent with the prediction that most second-order
basins should exceed this range, whereas most first-order basins
should not. The match is slightly better for the Allegheny Plateau,
whereas the gap between the modes occurs at approximately
300 , Pe , 350 for Gabilan Mesa, but this difference is small compared with the scale discrepancy in Fig. 1. In addition, most first-order
basins in both landscapes exceed the critical value of Pe < 60 for
the development of incipient valleys (Figs 3 and 4). Thus, drainage
0
Wavelength
Slope length
Perturbation
grows
−0.5
−1.0
0.1
Normalized growth rate
1
+0.5
Perturbation
decays
−1.5
20
60
100
Pe
140
180
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Figure 3 | Growth rates of incipient valleys on
inclined slopes. Left, wire mesh plots showing
vertically exaggerated examples of sinusoidal
perturbations on inclined, planar slopes. Right,
normalized growth rate (colour coded) of
sinusoidal perturbations as a function of Péclet
number, Pe, and aspect ratio (wavelength divided
by slope length). Normalized growth rate is
proportional to the fractional rate of change of the
standard deviation of surface elevation after the
background slope has been removed. Contour
interval is 0.05, and the black line is the zero
contour. Inset, illustration of the hypothetical
position of such a slope within a larger drainage
basin.
LETTER RESEARCH
a
h
t
row
ey g
Vall bility
a
t
ins
Full Methods and any associated references are available in the online version of
the paper.
ing
nch
Bra bility
a
t
ins
Received 16 April; accepted 27 September 2012.
Allegheny
Plateau
1.
2.
3.
4.
b
5.
Gabilan
Mesa
6.
7.
8.
10
102
103
104
9.
Pe
10.
Figure 4 | Péclet number distributions of first- and second-order drainage
basins in the study sites. a, Allegheny Plateau; b, Gabilan Mesa. Coloured lines
are histograms with counts weighted by drainage basin area to compensate for
the greater abundance of smaller basins. Inset sketches are examples of firstand second-order basins. Grey bars indicate critical values of Pe for valley
growth (Fig. 3) and branching (Fig. 2). Basins analysed are the same as in Fig. 1.
11.
12.
13.
14.
non-exclusive explanations for the difference in channel incision: first,
the weaker rocks at Gabilan Mesa are easier to erode; and second,
highly seasonal rainfall and sparser vegetation at Gabilan Mesa promote runoff and inhibit infiltration, such that larger surface flows occur
at a given drainage area and mean rainfall rate. The broader implication of this result is that the finely branched tips of river networks,
which form both the skeleton and the circulatory system of Earth’s
landscapes, carry a fundamental signature of rock strength, climate and
life. A remaining challenge is to further quantify this signature by
relating specific materials, mechanisms and conditions to the rate constants used to describe landscape evolution over geologic time.
15.
16.
17.
18.
19.
20.
21.
22.
23.
METHODS SUMMARY
Topographic analysis. We identified valleys in laser altimetry data as areas of
anomalously positive contour curvature (Supplementary Fig. 1). We then traced
valley networks over a larger region using a steepest-descent flow routing procedure and a drainage area threshold based on the curvature criterion, and assigned
Strahler stream orders to these networks (Supplementary Fig. 2). Drainage basin
areas, A, are determined just upstream of the junction
pffiffiffiffiffiffiwith a higher-order link. We
approximated22 drainage basin1 length as L < 3A, and calculated the Péclet
mz2
number as Pe~ðK=DÞð3AÞ
.
Landscape evolution model. The model21,23 solves an equation for the time
evolution of an elevation surface due to rock uplift or boundary lowering, channel
incision and downslope soil transport. We solved this equation forward in time
with a finite difference method23 on a rectangular grid with periodic x boundaries,
lowering y boundaries, and a low-relief, randomly rough initial surface. We varied
the Péclet number by varying K/D and performed 1,600 runs with different initial
conditions to determine the probability densities in Fig. 2a.
Cosmogenic nuclides. We estimated long-term erosion rates by measuring the
concentration of cosmogenic 10Be in quartz grains in stream sediment (Supplementary Table 1) and converting these concentrations to surface erosion rates
based on rates of 10Be production and decay.
Calculation of K and D. We assumed steady state hillslope topography and
calculated the soil transport coefficient as D 5 –E/+2 zh , where E is the surface
erosion rate determined from cosmogenic 10Be and +2 zh is the Laplacian of
elevation on hilltops, which has been measured from laser altimetry21 (Supplementary Table 2). The channel incision coefficient K was calculated by dividing
D by previous estimates21 of D/K, and the channel incision coefficient for a ref21
erence drainage area, KAm
ref , was calculated with previous estimates of m and
Aref 5 104 m2 (Supplementary Table 2).
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Durham, D. L. Geology of the southern Salinas Valley area, California. Prof. Pap. US
Geol. Surv. 819, 1–111 (1974).
Perron, J. T., Kirchner, J. W. & Dietrich, W. E. Formation of evenly spaced ridges and
valleys. Nature 460, 502–505 (2009).
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Bonnet, S. Shrinking and splitting of drainage basins in orogenic landscapes from
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Supplementary Information is available in the online version of the paper.
Acknowledgements This study was supported by the US National Science Foundation
Geomorphology and Land Use Dynamics programme through award EAR-0951672 to
J.T.P. and by the US Department of Defense through a National Defense Science and
Engineering Graduate Fellowship to P.W.R. J.T.P. is a Scholar in the Canadian Institute
for Advanced Research (CIFAR). The authors thank T. Clifton and G. Chmiel for
assistance with sample preparation, and the Orradre family of San Ardo, California, and
numerous landowners in Greene County, Pennsylvania, for granting access to their land.
Author Contributions J.T.P. conceived of the study, performed the numerical
modelling, and wrote the paper. J.T.P., P.W.R. and K.L.F. conducted the fieldwork. P.W.R.
processed the 10Be samples, and P.W.R. and K.L.F. analysed the 10Be data. J.T.P. and
M.L. performed the topographic analyses.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Readers are welcome to comment on the online version of the paper. Correspondence
and requests for materials should be addressed to J.T.P. ([email protected]).
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RESEARCH LETTER
METHODS
Topographic analysis. We mapped valley networks in the central Gabilan Mesa
(GM) in California, between the towns of King City and San Miguel, and the
Waynesburg Hills area of the Allegheny Plateau (AP) in Pennsylvania and West
Virginia, in the Dunkard Creek, Fish Creek and South Fork Ten Mile Creek watersheds, using a combination of airborne laser altimetry and the US National
Elevation Dataset (NED). Laser altimetry for GM was acquired and processed
by the National Center for Airborne Laser Mapping (NCALM), and laser altimetry
for AP was produced by the PAMAP program of the Pennsylvania Department of
Conservation and Natural Resources. Both laser altimetry data sets were filtered to
remove vegetation and gridded to a horizontal point spacing of 1 m.
We first used the laser altimetry to map valley networks over subregions of
roughly 25 km2. We mapped valleys rather than channels to focus on the longterm topographic signature of channel incision rather than the most recent episodes (particularly since the latter may have been influenced by human land use).
To objectively map the valley network, we used a procedure that identifies valleys
as areas with anomalously convergent topography, as defined below, and then uses
flow routing to bridge any gaps in the resulting map. The steps in this procedure
are illustrated in Supplementary Fig. 1. First, elevations were smoothed with a
Gaussian filter with a standard deviation of 10 m (GM) or 20 m (AP) to reduce
fine-scale roughness (Supplementary Fig. 1a). Contour curvature (Supplementary
Fig. 1b) was then calculated as30,31:
2 2
2 2
Lz L z
Lz Lz L2 z
Lz L z
{2
z
Ly Lx2
Lx Ly LxLy
Lx Ly2
ð1Þ
kc ~
2 2 !32
Lz
Lz
z
Lx
Ly
Following a recently proposed procedure32,33, we identified valleys as areas in
which contour curvature is sufficiently positive that it deviates from the approximately Gaussian frequency distribution of curvature over most of the terrain.
Specifically, we defined valleys as clusters of points with kc greater than two
standard normal quantiles above the mean, which is the value at which the frequency distribution of kc begins to deviate substantially from a Gaussian distribution (Supplementary Fig. 1c). To exclude small, scattered local minima on
hillslopes that satisfy the curvature criterion, we additionally required that the
clusters be at least 10 m2 in size. To exclude depressions in nearly planar topography (such as on alluviated valley floors) that are not part of the valley network,
we required that clusters include at least one point with a drainage area (calculated
with the D-infinity algorithm34) greater than two standard normal quantiles above
the mean.
Applying the curvature criterion produced a binary map of pixel clusters that lie
within areas of strongly convergent topography (Supplementary Fig. 1d). We then
routed flow over the entire landscape using a steepest-descent algorithm35, identified the pixel in each cluster with the largest drainage area, and traced flow paths
downslope from these pixels to the boundaries of the elevation grid. The resulting
network of clusters joined by flow paths (Supplementary Fig. 1e) contained some
parallel flow paths in areas with subtly varying slope directions, because the
steepest-descent algorithm is insensitive to changes in flow direction of less than
45u. Parallel flow paths were merged (Supplementary Fig. 1f) by applying a morphological closing operation36 with a disk-shaped structuring element with a
radius of 10 m. The resulting skeleton was thinned to a network one pixel wide
and pruned to remove spurs less than 10 m long (Supplementary Fig. 1g).
We then used this laser altimetry-based drainage network to calibrate a procedure for mapping the drainage network over a much larger area using coarser
elevation data. In this procedure, the steepest-descent flow-routing algorithm was
applied to the 10 m resolution (GM) or 30 m resolution (AP) NED over the same
area covered by the laser altimetry, the drainage network was identified as the set of
points with drainage area greater than a minimum value, Amin, and the network
was pruned to remove spurs less than 20 m (GM) or 30 m (AP) in length. The laser
altimetry-based network was then compared with the NED-based network.
For each study site, we identified the value of Amin that maximized the number
of valleys appearing as first-order links in both networks, and found Amin 5
10,000 m2 for GM and 50,400 m2 for AP. Using these values, we mapped the
drainage networks over larger geographic areas with the NED, increasing our
sample size from tens of drainage basins to several thousand, and assigned
Strahler stream orders11,16 to the networks. For each first-order or second-order
link in a network, we identified the associated drainage basin as the area that drains
to the point on the link just upstream of the junction with a higher-order link.
Comparison of the laser altimetry-based valley network with NED-based drainage
basin boundaries (Supplementary Fig. 2) indicates that the calibrated analysis of
NED elevation data correctly identifies the boundaries of most first-order and
second-order drainage basins, despite the coarser resolution. The mapped valley
network corresponds to areas that have experienced significant channel incision
over geologic time, but does not necessarily coincide with the extent of the currently active channel network37. Recent episodes of channel incision may extend
further upslope than the long-term average in some locations (such as the small
gullies visible in Supplementary Fig. 2), while in others, valleys that have experienced significant long-term incision may lack currently active channels.
To transform drainage basin areas into the lengths, L, used to calculate the
Péclet number, Pe, we used the well-known observation that drainage basin area
scales with the square of basin length to a good approximation
pffiffiffiffiffiffi over many orderspofffiffiffi
magnitude in basin size38. It has been found22 that L < 3A, with the factor of 3
describing the non-square shape of drainage basins.
Thus, for a drainage basin
1
with an area A, we calculate Pe~ðK=DÞð3AÞmz2 . We used drainage area to
calculate equivalent basin lengths rather than measuring basin length directly
because drainage area is an integrated measure of basin size that is less sensitive
to differences in basin shape.
In addition to this quantitative test of the proposed mechanisms for the
development of tributary networks, visual inspection of the topography provides
a qualitative means of identifying evidence that these mechanisms have operated
in a landscape. For example, Supplementary Fig. 3 shows several adjacent drainage
basins that appear to have undergone the instabilities illustrated in Figs 2c and 3.
Landscape evolution model. The landscape evolution model is a modified version of the Tadpole model21,23, which solves an equation for the time evolution of
an elevation surface, z(x, y), derived13,23,39 from conservation of mass and rate laws
for soil creep and channel incision,
Lz
ð2Þ
~D+2 z{KAm j+zjzE
Lt
where z is elevation, t is time, D is the soil transport coefficient, A is drainage area,
K and m are constants, and E is the rate of boundary lowering. Elevation is
measured relative to the lowering boundary, such that E appears in equation (2)
as a surface uplift rate. The linear diffusion term is based on the theoretical
expectation40,41, supported by field measurements42–47, that soil creep flux on hillslopes with gentle to moderate gradients is driven by isotropic disturbance (due to
animal burrowing, for example) and gravitational settling, such that soil flux is
proportional to the topographic gradient. The nonlinear kinematic wave term is
based on studies of erosional channels28,48–50 suggesting that channel incision rate
is proportional to the rate of energy expenditure by the flow of water51, or ‘stream
power’, per unit area of the bed under normal flow. A term with the same form—
that is, a power law dependence of channel incision rate on drainage area and
slope—is obtained if incision rate is instead assumed to be proportional to shear
stress52.
The linear diffusion term describes the smoothing effect of slope-dependent soil
creep, and the nonlinear kinematic wave term describes how changes in elevation
at the boundaries propagate upslope through the landscape, including a strong
positive feedback in valley incision through the dependence on drainage area. The
boundary lowering (or, equivalently, uplift relative to the boundaries), E, drives the
development of the landscape by steepening the topography. The competition
between these effects gives rise to a landscape with smooth, concave down ridgelines and sharper, concave-up valleys. As noted in the main text, the Péclet number
describes the relative magnitudes of the soil creep and channel incision terms23.
The use of a continuum model with constant coefficients implicitly averages in
space and time over many erosion and transport events, which would be difficult
to describe individually. For example, this model assumes that channel incision
and soil creep can occur in the same location, in a time-averaged sense, and is
therefore appropriate for low-order drainage basins in which episodes of channelization are interspersed with periods of diffusive infilling (Supplementary Fig. 4).
The modelling approach employed here makes it possible to explore the main
controls on landscape form in terms of a few parameters.
Equation (2) was solved numerically on a 300 3 100 (Nx 3 Ny) regular grid with
Dx, Dy 5 5 m using a finite difference method23. The grid was subject to a constant
lowering of the y boundaries intended to mimic a set of parallel, incising channels.
The x boundaries were periodic, such that the topography repeats with a period
equal to the length of the domain in the x direction. Experiments began with an
initial surface of uncorrelated Gaussian noise with a variance of 1 m2, and continued until the topography reached a dynamic equilibrium in which all points
were lowering at the same rate. In each experiment, incipient drainage basins
formed at the lowering y boundaries and advanced towards the interior of the
grid in a direction perpendicular to the y boundaries. A central drainage divide
formed where the basins advancing in opposite directions met roughly halfway
between the y boundaries. Supplementary Fig. 5 shows steady-state solutions for
three experiments.
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LETTER RESEARCH
We varied the Péclet number in the model by setting m 5 0.5 and changing the
ratio K/D. The length scale L was defined as half of the length of the domain in the y
direction, which is the approximate length of drainage basins that extend from the
central drainage divide to one of the lowering boundaries. In each model solution,
we identified drainage basin outlets as points adjacent to one of the lowering
boundaries with +2 z $ 0.005 m21, measured the drainage area at each outlet,
and normalized the areas by L2. While the mean characteristics of drainage basins
at steady state for a given Péclet number are independent of initial conditions, the
characteristics of individual basins are sensitive to initial conditions, and vary
about the mean53. To better characterize the size distribution of drainage basins
for a given Pe, we performed an ensemble of 1,600 simulations with identical
parameters and boundary conditions, but different initial surfaces. A probability
density map of the resulting population of more than 60,000 drainage basins as a
function of drainage basin size and Péclet number reveals a transition from a
unimodal to a bimodal size distribution over the interval 250 , Pe , 300 (Fig. 2).
Cosmogenic nuclides. We estimated long-term erosion rates in AP and GM by
measuring the concentration of cosmogenic 10Be in quartz grains in stream sediment, following an established procedure54. This yields erosion rates that are
spatially averaged over the sampled drainage basins and temporally averaged over
the characteristic time of 10Be accumulation in the sampled quartz, which is
typically 103–105 yr in most eroding landscapes. We collected sediment samples
from three drainage basins in AP and two in GM in 2009 and 2010. All sampled
basins have quartz-rich lithologies (sandstone in AP, sandstone and quartzite
conglomerate in GM). We separated and dissolved quartz from each sample
following a standard procedure55 and isolated 10Be using cation exchange56.
10
Be/9Be ratios were measured with Accelerator Mass Spectrometry at PRIME
Lab, Purdue University, in February 2011. The mean location and elevation of
each drainage basin were used to determine 10Be production rates57. For
small drainage basins like those analysed here, the uncertainty introduced by using
a single location and elevation to approximate the production rate is typically
much smaller than uncertainties associated with other assumptions required to
calculate basin-averaged erosion rates57. A correction factor accounting for topographic shielding from cosmic rays57 was calculated at each pixel in a 4 m per pixel
version of the laser altimetry map of GM, and the mean value for each drainage
basin was used. The shielding correction at all of the AP sites was found to
be negligible.
Because the sediments that make up GM are young, and might therefore have
been exposed at the surface recently, we corrected the 10Be concentrations in the
GM samples by subtracting the inherited concentration measured in three
shielded samples with the same lithology. The shielded samples were collected
in 2002 from recently excavated drill platforms near the 2009–10 surface sample
locations. When the samples were originally buried to depths much deeper than
1 m, they were shielded from subsequent production and the 10Be concentration
was altered only by radioactive decay. The shielded samples were processed at the
University of California, Berkeley, and 10Be/9Be ratios were measured with
Accelerator Mass Spectrometry at Lawrence Livermore National Laboratory in
2005. The mean 10Be concentration of the shielded samples was subtracted from
the 10Be concentrations of the surface samples to determine the 10Be concentration
accumulated during recent exposure (Supplementary Table 1). Given the close
vertical proximity of the shielded samples and the 1.36 Myr half-life of 10Be, we
interpret the difference in 10Be concentration in the shielded samples to reflect
slightly different source and transport histories rather than different durations
of burial.
10
Be concentrations of the surface samples at both sites were converted into
erosion rates using the CRONUS calculator57, version 2.2. Measured quantities
used in this calculation, and the resulting erosion rates, are summarized in
Supplementary Table 1. The uniform relief of both landscapes and the consistency
of the erosion rates among sampling locations suggest that these rates are representative of the surrounding landscape. We therefore average the erosion rates to
arrive at a single rate for each landscape: 92 6 5 t km22 yr21 for AP, and 206 6 27 t
km22 yr21 for GM. The uncertainties were taken to be the larger of the standard
error of the mean of the site-specific rates and the pooled site-specific uncertainties.
For comparison with the model predictions, these mass removal rates must be
converted to surface erosion rates with units of [L T21]. Dividing by measured
soil densities of 1,660 6 110 kg m23 in AP and 1,400 6 100 kg m23 in GM yields
surface erosion rates of 55 6 5 m Myr21 in AP and 147 6 22 m Myr21 in GM. This
calculation assumes that chemical mass removal by dissolution is negligible.
Calculation of K and D. At steady state (hz/ht 5 0), the surface erosion rate at any
location equals the boundary lowering rate, and equation (2) reduces to:
{E~D+2 z{KAm j+z j
On hilltops, where drainage area and slope are small, Am j+z j<0, and
ð3Þ
D~{
E
+2 zh
ð4Þ
where +2 zh is the Laplacian of elevation on hilltops. We used equation (4) to
estimate D for AP and GM, using the surface erosion rates measured with cosmogenic nuclides as an estimate of E, and the hilltop Laplacian measured in ref. 21
(Supplementary Table 2). The uniform value of the Laplacian over each hillslope
supports the steady-state assumption in equation (3). Given D and an estimate of
D/K (ref. 21), it is straightforward to calculate K. The units of K depend on the
value of m. To facilitate comparisons among different landscapes where m can
vary, we compare the quantity KAm
ref , where Aref is a reference drainage area, here
taken to be 104 m2. The estimated m values for AP and GM are very similar
(Supplementary Table 2), so the difference in KAm
ref is due almost entirely to the
difference in K. Note that KAm
ref is faster than the actual channel incision rate,
which also depends on slope.
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